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<h1>Compensation</h1>

<p>There are two physical effects that affect the pitch of a vibrating string:</p>

<ul>
<li>String stiffness</li>
<li>String tension</li>
</ul>

<p>1. A string is stiff. That means that the effective length of a vibrating string, fixed and held into position at the ends, is shorter than the distance between the fixing points. In other words: The stiffness of the string effectively reduces the scale length.</p>

<p>To compensate this effect, the fixing points must be moved apart a little bit.</p>

<p>Different strings are different in stiffness. This means that each string needs a different amount of compensation.</p>

<p>The stiffness effect is determined by the <em>core</em> diameter of wound strings (and its stiffness, of course). That is the reason why a wound D (4th) string needs less bridge compensation than the unwound G (3rd) string.</p>

<p>2. When a string is pressed down, it is stretched. Stretching means increased tension, which in turn raises the pitch.</p>

<p>3. According to Greg Byers on <a href="http://www.byersguitars.com/Research/Research.html">www.byersguitars.com</a>, the harmonic overtones of a string are somewhat sharp, compared to the fundamental. This effect increases with harmonic order, and is also more prominent in the upper region of the fretboard. The effective result to a listener is that the notes souned increasingly sharp when going up the fretboard.</p>

<p>Fretting in the middle of the length requires (much) less force than fretting near the nut. This means that the string is stretched (much?) more on the first few frets.</p>

<p>A low action at the first fret minimizes the problem, but does not zero it, regardless that the string action at fret 12 is significantly higher (2.5 to 4 mm).</p>

<p>The same effect comes into play with frets above 12. These frets are sharpened by the effect.</p>

<p>3. A string can vibrate more freely over a fret than at the nut: While the string is pinched firmly on 3 sides, it can twist around over a fret, more or less freely, when press at a distance behind the fret. Additionally, the finger is much more flexible than the nut. As a result, the effective suspension point is farther away from the nut than from a fret. This effect would make all frets sound flat.</p>

<p>4. The sound energy in a string is proportional to the vibration amplitude, and also proportional to the frequency. That means that the amplitute of lower tones is somewhat lower than that of higher tones. Combined with the stiffness effect, this means that the effective suspension point is nearer to the higher frets than to the lower frets. As a result, higer frets sound flat, lower frets sound sharp.</p>

<p>5. The fretboard is not flat. On acoustic and classical guitars, the fretboard often recedes back in the body area, adding more string action height to the frets above 12 (or 14 on acoustic steel string guitars).</p>

<p>6. Different string types (nylon, metal-wound, materials) have different characteristics. So with different manufacturers. Hopefully, the same brand and type of strings is constistent over time.</p>

<p>7. Homogenity in mass distribution along the string length. The string length to pitch ratio is only valid for completely homogenous strings. New strings should be perfectly so. But strings wear by playing and being bent and shifted over the frets, so this is a factor. Influence on intonation? Definitively not a good...</p>

<p>8. Material deterioration. With new strings, the rate seems to be higher. New strings need frequent retuning. This should have no effect on intonation.</p>

<p>9. Finger pressure: Different players fret with different forces. Higher pressure makes the notes sharp. </p>

<p>10. Shape of fingers. I consider this factor rather low. But big ears might hear the difference.</p>

<p>11. Inherent bending. When a chord is fretted, the fingers do bend strings, depending on chord shape, style, hand geometry and strength of the artist. This influence is somewhat randomly, and thus must be rated individually.</p>

<p>12. Fret wear. Not only the height of the fret decreases, but the front edge moves towards the bridge a little bit. Both factors sharpen the note.</p>

<p>When I look at the scale length compensation tables, it comes to mind that a string setup must be compensated at the bridge saddle <em>and</em> the nut.</p>

<h2>Daniel Haines</h2>

<p>William Cumpiano wrote in his <a href="http://www.cumpiano.com/Home/Newsletters/Issues/newsletter%2019.htm">newsletter 19</a> about Daniel Haines, a researcher who specified the order of influenes:</p>

<ul>
<li>The compensation is directly proprotional to the square of the string clearance over the frets.</li>
<li>Directly proportional to the modulus of elasticicty (stiffness) of the string core material</li>
<li>Directly proportional to the cross-sectional area of the string core.</li>
<li>Inversely proportional to the open string length.</li>
<li>Inversley proportional to the string tension.</li>
</ul>

<h2>Practical Compensation Values</h2>

<p>Cumpiano, classical guitar: 0.1" (2.54 mm) for the usual range of scale lengths.</p>
<p>Cumpiano, steel string acoustic: 0.15" plus 1/8" slope in 3" on the bass side, for a scale length of 25" to 25 3/4".</p>

<table class="boxborders" style="border-collapse: collapse; table-layout: auto;"><caption>Rick Middleton's Compensation Table</caption>
<thead><tr>
	<th>String</th>
	<th>Note</th>
	<th>Compensation</th>
</tr></thead><tbody><tr>
	<td>1</td>
	<td>E4</td>
	<td>2.5</td>
</tr><tr>
	<td>2</td>
	<td>B3</td>
	<td>3</td>
</tr><tr>
	<td>3</td>
	<td>G3</td>
	<td>4</td>
</tr><tr>
	<td>4</td>
	<td>D3</td>
	<td>3</td>
</tr><tr>
	<td>5</td>
	<td>A2</td>
	<td>3.5</td>
</tr><tr>
	<td>6</td>
	<td>E2</td>
	<td>4</td>
</tr></tbody></table>

<h2>Links</h2>

<dl>
<dt><a href="http://www.mimf.com/nutcomp/">The Compensated Nut</a>; Stephen Delft.</dt>

<dt><a href="http://liutaiomottola.com/formulae/compensation.htm">String Compensation Calculator</a> by Liutaio Mottola</dt>
<dd>Based on investigations of Sjaak Elmendorp, publised in American Lutherie #104.</dd>

<dt><a href="http://setitupbetter.com/">Set It Up Better</a> web site</dt>
<dd>A rather confusing introduction to the matter, but good tips and tricks.</dd>

<dt><a href="http://wolflehmanngitarren.de/nowomensur/index.htm">Wolf Lehmann</a></dt>

<dt><a href="http://www.pjguitar.com/website/forums/articles/intonation.html">PJ Guitar</a></dt>

</dl>

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